So the bottom line is that although waves have interesting behavior there properties are not surprising.
We now make a giant leap and require that waves move from the phenomenological or perhaps derivable to the fundamental.
We require the basic elements in our theory, particle and interactions, to have characteristics similar to particles and characteristics similar to waves. These properties are simply “the way it is”. We will take the perspective that it is not justifiable in terms of some overriding principle merely valid based on observation. Taking this point of view is verified in practice to an almost undeniable degree. Quantum mechanics, the theory that merges these features successfully has very few doubters and stands an very firm ground.
Particle nature: BULLETS
- observations reveal that the particle of the theory always show up in the measurement as distinct chunks. You will never measure 1/2 of an electron or a partial photon. You will always get either one or none! To understand this we will need to be clear about the measurement process. It will be important to make predictions based on what you measure.
Wave nature: Interference
- The classic way to begin to understand the wave nature of QM is the interference particles and waves impinging on two slits. There will be an interference pattern in both cases. Waves from the classical view point contribute an amplitude
There are two types of idealized problems that are usually introduced to provide examples of quantum behavior.
Double slit accumulation
Our goal is to obtain an overview of the essential ingredients in QM. Here are some salient aspects:
To determine the probability of an event one squares the amplitude.
Familiar vector space of 3-d does not typically use the notion of a dual space when introducing the inner or dot product. However 4-d spacetime can be conveniently cast into the above form and the minus sign for the time component, when calculating length, can be introduced by allowing the space and its dual to have opposite signs for the t-component.
These are somewhat formal mathematical rules but they are essential. If one can learn to formulate QM as vector space with amplitudes and bases, then the behavior of QS can be extracted and a certain intuition can be built.
Consider the earth orbiting around the sun. We know the interaction and can solve for the orbit.
From the force you calculate the earth’s orbit and with initial condition you know where it is at all times.
Given the Hamitonian H one can calculate the quantum wave function that is equivalent to the full QS.
Know how the atom will behave in terms of the wave function.
Suppose the problem is too difficult to solve because the interaction is more complicated.
For the two slit experiment you propagate a particle to slit 1 (the interaction) and then to the screen.
For this I can write down an amplitude to reach the screen by nature of the interaction.
One difference in QM is that I need to find all possible ways to reach the ending state
I can imagine a more complex system with a series of interactions.
My goal would be to calculate the amplitude to travel along any given path and then sum over all possible paths. Feynman diagrams diagrammatically represent this deeply complicated mathematical process. A first order diagram in some sense shows the interaction as a single step process, as if you can get a good result by correcting the trajectory one time classically. High order diagrams are similar to the multiple corrections required to find the asteroids final trajectory. In addition to the expansion of the interaction Feynman diagrams represent the multitude of ways that quantum systems can travel. The amplitudes for each trajectory will be included and interference may result.
There are several aspects of QM we will need to understand as general concepts.
Quantum states form a vector space (Hilbert space).
Operators may represent observables so their behavior and interpretation tells us how to think about particle properties.